M1L7u.txt

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OK.
We can get started with a very short chapter, which
is about g2, the g2 measurement for light and atoms.
I don't think you're going to find the discussion I'm going
to present you in any textbook.
It is about whether g2 is 1 or 2, whether we have fluctuations
or not.
And the discussion will be whether g2 of 2 and g2 of 1
are quantum effect or classical effects.
So I want to give you, here in this discussion,
four different derivations of whether g2 is 1 or g2 is 2.
And they look very, very different.
Some are based on classical physics.
Some are based on the concept of interference.
And some are based on the quantum indistinguishability
of particles.
And once you see all those four different explanations,
I think you see the whole picture,
and I hope you understand something.
So again, it's a long story about factors of 2,
but there are some factors of 2 which are purely calculational,
and there are other factors of 2 which involve
a hell of a lot of physics.
I mean, there are people who say the g2 factor of 2
is really the difference between ordinary light and laser light.
For light from a light bulb, g-2 is 2.
For light from a laser, g2 is 1.
And this is the only fundamental difference between laser light
and ordinary light.
So this factor of 2 is important.
And I want to, therefore, have this additional discussion
of the g2 function.
So let me remind you that g2 of 0
is the normalized probability to detect two photons, or two
particles, simultaneously.
And so far, we have discussed it for light.
And the results we obtained by using our quantum formulation
of light with creation-annihilation operator,
we found that g2 is 2.
In the situation that we had blackbody radiation, which
we can call thermal light-- it sometimes
goes by the name chaotic light.
Sometimes it's called classic light.
But that may be a misnomer, because I regard the laser
beam as a very classical form of light.

This is sometimes called bunching,
because 2 is larger than 1, so pairs of photons
appear bunched up.
You have a higher probability than you would naively
expect of detecting two photons simultaneously.
And then we had the situation of laser light and coherent light,
where the g2 function was 1.
And I want to shed some light on those two cases.
We have discussed the extreme case of a single photon, where
the probability of detecting two photons
is 0 for trivial reasons.
So you have a g2 function of 0.
But this is not what I want to discuss here.
I want to shed some light on when do we
get a g2 function of 1, when do we get a g2 function of 2?
And one question we want to address when we've
achieved the g2 function of 2, is this
a classical or quantum effect?
Do you have an opinion?
Who thinks-- you have question?
In the homework problem, where we
had the linear superposition, like alpha and minus alpha,
and an alpha and plus alpha, we found
that one can have a g2 grade and one can have g2 less than 1.
So maybe g2 isn't a great discriminator of [INAUDIBLE]
quantum or very classic.

That's all gestalt, ja?
You may be right.
It's-- let's come back to that.
I think that's one opinion.
The g2 function may not be a discriminator,
because we can have g2 of 1 and g2 of 2 purely classical.
But why classical light behaves classically,
maybe that's what we can understand there.
And maybe what I want to tell you is that a lot of properties
of classical light can be traced down
to the indistinguishability of bosons which are photons.
So in other words, we shouldn't be surprised
that something which seems purely classical
is deeply rooted in quantum physics.
But I'm ahead of my agenda.
So let me start now.
I want to offer you four different views.

And the first one is that we have random intensity
fluctuations.
Think of a classical light source.
And we assume that if things are really random,
they are described by a Gaussian distribution.

And if you switch on a light bulb, what
you get if you measure the intensity, when you measure
what is the probability that the momentary intensity is I,
well, you have to normalize it with the average intensity.
But what you get is pretty much an exponential distribution.
And this exponential distribution
has a maximum at I equals 0.
So the most probable intensity of all intensities
when you switch on a light bulb is
that you have zero intensity at a given moment.
But the average intensity is by average.
So you can easily, for such a distribution,
for such an exponential distribution,
figure out what is the average of I to the power n.
It is related to I average to the power n,
but it has an n factorial.
And what is important is the case for our discussion of n
equals 2, where the square of the intensity
averaged is 2 times the average intensity squared.
And classically, g2 is the probability
of detecting two photons simultaneously,
which is proportional to I square.
We have to normalize it, and we normalize it
by I average square.
And this gives 2.
So simply, light with Gaussian fluctuations
would give rise to a g2 function of 2.
Since it's random fluctuations, it's also called chaotic light.
And the physical picture is the following.
If you detect a photon, the light is fluctuating,
but whenever you detect the first photon,
it is more probable that you detect the first photon
when the intensity happened to be high.
And then since the intensity is high,
the probability for the next photon
is higher than the average probability.

So therefore, you get necessarily a g2 function of 2.
So this is the physics of it.
So let me just write that down.
The first photon is more likely to be
detected when the intensity fluctuations give
high intensity.
And then we get this result.
Let us discuss a second classical view, which
I can call wave interference.
If you have-- and this is really important.
A lot of people get confused about it.
If you have light in only one mode, this would be the laser,
or for atoms, the Bose-Einstein condensate.

One mode means a single wave, so if we have plain waves,
we can describe all the photons or all the atoms
by this wave function.

So what is a g2 function for an object like this?

One.
Trivially one.
Because if something is a plain wave, a single wave,
all correlation functions factorize.
You have a situation that I to the power n average, this I
average to the power n.
And that means that gn is 1 for all n.
OK.
But let's now assume that we have two.
We can also use more, but I want to restrict it to two.
That we have only two modes.
And two modes can interfere.
So let me apply it to those two modes
a simple model-- whether it's simple or not is relative.
So it goes like as follows.
If you have two modes, both of unity intensity,
then the average intensity is 2.
But if you have interference, then the normalized intensity
will vary between 0 and 4.
Constructive interference means you get twice as much
as average.
Destructive means you get nothing.
So therefore, the entity squared will vary between 0 and 16.
So if I just used the two extremes, it works out well.
You have an I square, which is 8,
the average of constructive interference
and destructive interference.
And this is 2 times the average intensity, which was 2 squared.
So therefore, if we simply allow fluctuations
into the interference of two modes,
we find that the g2 function is 2.

So this demonstrates that g2 of 2
has its deep origin in wave interference.
And indeed, if you take a light bulb which emits photons,
you have many, many atoms in your tungsten filament
which can emit.
They emit waves.
And since they have different positions,
the waves arrive at your detector with random phases.
And if you really write that down in the model--
this is nicely done in the book by Loudon--
you realize that random interference
between waves results in an exponential distribution
of intensity.
So most people wouldn't make the connection,
but there is a deep and fundamental relationship
between random interference and the most random distribution,
the exponential distribution intensity which characterizes
thermal light or chaotic light.
So let me just write that down.
So the Gaussian intensity distribution--
actually it's an exponential intensity distribution,
but if you write the intensity distribution,
it's a distribution in the electric field,
intensity becomes e square, then it becomes a Gaussian, but.
So Gaussian exponential intensity distribution
in view number one is indeed the result of interference.

Any questions?
So these are the two classical views.